Functional analytic properties and regularity of the Möbius-invariant Willmore flow in $${\mathbf {R}}^n$$ R n

In this article we continue the author’s investigation of the Möbius-invariant Willmore flow moving parametrizations of umbilic-free tori in $${\mathbf {R}}^n$$ R n and in the n-sphere $${\mathbf {S}}^n$$ S n . In the main theorems of this article we prove basic properties of the evolution operator of the “DeTurck modification” of the Möbius-invariant Willmore flow and of its Fréchet derivative by means of a combination of the author’s results about this topic with the theory of bounded $${\mathcal {H}}_{\infty }$$ H ∞ -calculus for linear elliptic operators due to Amann, Denk, Duong, Hieber, Prüss and Simonett with Amann’s and Lunardi’s work on semigroups and interpolation theory. Precisely, we prove local real analyticity of the evolution operator $$[F\mapsto {\mathcal {P}}^*(\,\cdot \,,0,F)]$$ [ F ↦ P ∗ ( · , 0 , F ) ] of the “DeTurck modification” of the Möbius-invariant Willmore flow in a small open ball in $$W^{4-\frac{4}{p},p}({\varSigma },{\mathbf {R}}^n)$$ W 4 - 4 p , p ( Σ , R n ) , for any $$p\in (3,\infty )$$ p ∈ ( 3 , ∞ ) , about any fixed smooth parametrization $$F_0:{\varSigma } \longrightarrow {\mathbf {R}}^n$$ F 0 : Σ ⟶ R n of a compact and umbilic-free torus in $${\mathbf {R}}^n$$ R n . We prove moreover that the entire maximal flow line $${\mathcal {P}}^*(\,\cdot \,,0,F_0)$$ P ∗ ( · , 0 , F 0 ) , starting to move in a smooth and umbilic-free initial immersion $$F_0$$ F 0 , is real analytic for positive times, and that therefore the Fréchet derivative $$D_{F}{\mathcal {P}}^*(\,\cdot \,,0,F_0)$$ D F P ∗ ( · , 0 , F 0 ) of the evolution operator in $$F_0$$ F 0 can be uniquely extended to a family of continuous linear operators $$G^{F_0}(t_2,t_1)$$ G F 0 ( t 2 , t 1 ) in $$L^p({\varSigma },{\mathbf {R}}^n)$$ L p ( Σ , R n ) , whose ranges are dense in $$L^{p}({\varSigma },{\mathbf {R}}^n)$$ L p ( Σ , R n ) , for every fixed pair of times $$t_2\ge t_1$$ t 2 ≥ t 1 within the interval of maximal existence $$(0,T_{{\mathrm{max}}}(F_0))$$ ( 0 , T max ( F 0 ) ) .